001/* 002 * Copyright (c) 2022, Oracle and/or its affiliates. All rights reserved. 003 * 004 * Licensed under the Apache License, Version 2.0 (the "License"); 005 * you may not use this file except in compliance with the License. 006 * You may obtain a copy of the License at 007 * 008 * http://www.apache.org/licenses/LICENSE-2.0 009 * 010 * Unless required by applicable law or agreed to in writing, software 011 * distributed under the License is distributed on an "AS IS" BASIS, 012 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express implied. 013 * See the License for the specific language governing permissions and 014 * limitations under the License. 015 */ 016 017/* 018 * Port of portions of the "Freely Distributable Math Library", version 5.3, 019 * from C to Java. This file ports the function e_lgamma_r and its helper 020 * function sinpi to produce Java functions GammaMath.lgamma and 021 * GammaMath.sinpi; 022 */ 023 024/* @(#)e_lgamma_r.c 1.3 95/01/18 025 * ==================================================== 026 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 027 * 028 * Developed at SunSoft, a Sun Microsystems, Inc. business. 029 * Permission to use, copy, modify, and distribute this 030 * software is freely granted, provided that this notice 031 * is preserved. 032 * ==================================================== 033 */ 034 035/* __ieee754_lgamma_r(x, signgamp) 036 * Reentrant version of the logarithm of the Gamma function 037 * with user provided pointer for the sign of Gamma(x). 038 * 039 * Method: 040 * 1. Argument Reduction for 0 < x <= 8 041 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 042 * reduce x to a number in [1.5,2.5] by 043 * lgamma(1+s) = log(s) + lgamma(s) 044 * for example, 045 * lgamma(7.3) = log(6.3) + lgamma(6.3) 046 * = log(6.3*5.3) + lgamma(5.3) 047 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 048 * 2. Polynomial approximation of lgamma around its 049 * minimum ymin=1.461632144968362245 to maintain monotonicity. 050 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 051 * Let z = x-ymin; 052 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 053 * where 054 * poly(z) is a 14 degree polynomial. 055 * 2. Rational approximation in the primary interval [2,3] 056 * We use the following approximation: 057 * s = x-2.0; 058 * lgamma(x) = 0.5*s + s*P(s)/Q(s) 059 * with accuracy 060 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 061 * Our algorithms are based on the following observation 062 * 063 * zeta(2)-1 2 zeta(3)-1 3 064 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 065 * 2 3 066 * 067 * where Euler = 0.5771... is the Euler constant, which is very 068 * close to 0.5. 069 * 070 * 3. For x>=8, we have 071 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 072 * (better formula: 073 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 074 * Let z = 1/x, then we approximation 075 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 076 * by 077 * 3 5 11 078 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 079 * where 080 * |w - f(z)| < 2**-58.74 081 * 082 * 4. For negative x, since (G is gamma function) 083 * -x*G(-x)*G(x) = pi/sin(pi*x), 084 * we have 085 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 086 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 087 * Hence, for x<0, signgam = sign(sin(pi*x)) and 088 * lgamma(x) = log(|Gamma(x)|) 089 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 090 * Note: one should avoid compute pi*(-x) directly in the 091 * computation of sin(pi*(-x)). 092 * 093 * 5. Special Cases 094 * lgamma(2+s) ~ s*(1-Euler) for tiny s 095 * lgamma(1)=lgamma(2)=0 096 * lgamma(x) ~ -log(x) for tiny x 097 * lgamma(0) = lgamma(inf) = inf 098 * lgamma(-integer) = +-inf 099 * 100 */ 101 102package org.tribuo.util.infotheory; 103 104/** 105 * Static functions for computing the Gamma and log Gamma functions on real valued inputs. 106 */ 107public final class Gamma { 108 private static final double 109 two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ 110 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 111 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 112 zero= 0.00000000000000000000e+00, 113 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 114 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 115 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 116 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 117 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 118 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 119 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 120 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 121 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 122 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 123 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 124 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 125 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 126 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 127 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 128 /* tt = -(tail of tf) */ 129 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 130 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 131 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 132 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 133 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 134 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 135 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 136 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 137 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 138 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 139 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 140 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 141 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 142 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 143 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 144 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 145 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 146 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 147 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 148 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 149 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 150 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 151 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 152 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 153 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 154 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 155 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 156 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 157 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 158 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 159 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 160 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 161 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 162 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 163 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 164 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 165 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 166 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 167 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 168 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 169 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 170 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 171 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 172 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 173 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 174 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 175 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 176 177 /** 178 * Private constructor to ensure that the class is never instantiated. 179 */ 180 private Gamma() {} 181 182 /** 183 * Return the low-order 32 bits of the double argument as an int. 184 * @param x The input double. 185 * @return The lower 32-bits as an int. 186 */ 187 private static int __LO(double x) { 188 return (int)Double.doubleToRawLongBits(x); 189 } 190 191 /** 192 * Return the high-order 32 bits of the double argument as an int. 193 * @param x The input double. 194 * @return The upper 32-bits as an int. 195 */ 196 private static int __HI(double x) { 197 return (int)(Double.doubleToRawLongBits(x) >> 32); 198 } 199 200 private static double sin_pi(double x) { 201 double y,z; 202 int n,ix; 203 204 ix = 0x7fffffff&__HI(x); 205 206 if (ix<0x3fd00000) { 207 return Math.sin(pi * x); 208 } 209 y = -x; /* x is assumed negative */ 210 211 /* 212 * argument reduction, make sure inexact flag not raised if input 213 * is an integer 214 */ 215 z = Math.floor(y); 216 if(z!=y) { /* inexact anyway */ 217 y *= 0.5; 218 y = 2.0*(y - Math.floor(y)); /* y = |x| mod 2.0 */ 219 n = (int) (y*4.0); 220 } else { 221 if(ix>=0x43400000) { 222 y = zero; 223 n = 0; /* y must be even */ 224 } else { 225 if(ix<0x43300000) 226 z = y+two52; /* exact */ 227 n = __LO(z)&1; /* lower word of z */ 228 y = n; 229 n<<= 2; 230 } 231 } 232 switch (n) { 233 case 0: 234 y = Math.sin(pi*y); 235 break; 236 case 1: 237 case 2: 238 y = Math.cos(pi*(0.5-y)); 239 break; 240 case 3: 241 case 4: 242 y = Math.sin(pi*(one-y)); 243 break; 244 case 5: 245 case 6: 246 y = -Math.cos(pi*(y-1.5)); 247 break; 248 default: 249 y = Math.sin(pi*(y-2.0)); 250 break; 251 } 252 return -y; 253 } 254 255 256 /** 257 * Function to calculate the log of a Gamma function. Negative integer values will return NaN. 258 * @param x The value to calculate for. 259 * @return The log of the Gamma function applied to x. 260 */ 261 public static double logGamma(double x) { 262 double t,y,z,nadj,p,p1,p2,p3,q,r,w; 263 int i,hx,lx,ix; 264 265 if((x <= 0) && (Math.floor(x) == x)) { 266 return Double.NaN; 267 } 268 269 hx = __HI(x); 270 lx = __LO(x); 271 nadj = zero; 272 273 /* purge off +-inf, NaN, +-0, and negative arguments */ 274 ix = hx&0x7fffffff; 275 if(ix>=0x7ff00000) { 276 return x * x; 277 } 278 if ((ix|lx)==0) { 279 return one / zero; 280 } 281 if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ 282 if(hx<0) { 283 return -Math.log(-x); 284 } else { 285 return -Math.log(x); 286 } 287 } 288 if(hx<0) { 289 if(ix>=0x43300000) { /* |x|>=2**52, must be -integer */ 290 return one / zero; 291 } 292 t = sin_pi(x); 293 if(t==zero) { 294 return one / zero; /* -integer */ 295 } 296 nadj = Math.log(pi/Math.abs(t*x)); 297 x = -x; 298 } 299 300 /* purge off 1 and 2 */ 301 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) { 302 r = 0; 303 /* for x < 2.0 */ 304 } else if(ix<0x40000000) { 305 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 306 r = -Math.log(x); 307 if(ix>=0x3FE76944) { 308 y = one-x; 309 i= 0; 310 } else if(ix>=0x3FCDA661) { 311 y= x-(tc-one); 312 i=1; 313 } else { 314 y = x; 315 i=2; 316 } 317 } else { 318 r = zero; 319 if(ix>=0x3FFBB4C3) { 320 y=2.0-x; 321 i=0; 322 } /* [1.7316,2] */ 323 else if(ix>=0x3FF3B4C4) { 324 y=x-tc; 325 i=1; 326 } /* [1.23,1.73] */ 327 else { 328 y=x-one; 329 i=2; 330 } 331 } 332 switch(i) { 333 case 0: 334 z = y*y; 335 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 336 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 337 p = y*p1+p2; 338 r += (p-0.5*y); 339 break; 340 case 1: 341 z = y*y; 342 w = z*y; 343 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 344 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 345 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 346 p = z*p1-(tt-w*(p2+y*p3)); 347 r += (tf + p); 348 break; 349 case 2: 350 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 351 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 352 r += (-0.5*y + p1/p2); 353 } 354 } else if(ix<0x40200000) { /* x < 8.0 */ 355 i = (int)x; 356 t = zero; 357 y = x-i; 358 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 359 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 360 r = half*y+p/q; 361 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 362 switch(i) { 363 case 7: 364 z *= (y+6.0); 365 //$FALL-THROUGH$ 366 case 6: 367 z *= (y+5.0); 368 //$FALL-THROUGH$ 369 case 5: 370 z *= (y+4.0); 371 //$FALL-THROUGH$ 372 case 4: 373 z *= (y+3.0); 374 //$FALL-THROUGH$ 375 case 3: 376 z *= (y+2.0); 377 //$FALL-THROUGH$ 378 r += Math.log(z); 379 break; 380 } 381 /* 8.0 <= x < 2**58 */ 382 } else if (ix < 0x43900000) { 383 t = Math.log(x); 384 z = one/x; 385 y = z*z; 386 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 387 r = (x-half)*(t-one)+w; 388 } else { 389 /* 2**58 <= x <= inf */ 390 r = x * (Math.log(x) - one); 391 } 392 if(hx<0) { 393 r = nadj - r; 394 } 395 return r; 396 } 397 398 /** 399 * Function to calculate the sign to be used for the result of a gamma function. 400 * @param x The value the gamma function is being constructed on. 401 * @return 1 or -1 depending on the sign that should be applied. The sign is then applied by multiplying by this value. 402 */ 403 private static int getSign(double x) { 404 if(x<0 && ((int)x)%2 == 0) { 405 return -1; 406 } else { 407 return 1; 408 } 409 } 410 411 /** 412 * Function to calculate the value of a Gamma function. Negative integer values will return NaN. 413 * @param x The value to calculate for. 414 * @return The value of the Gamma function applied to x. 415 */ 416 public static double gamma(double x) { 417 double r = logGamma(x); 418 int sign = getSign(x); 419 return sign * Math.exp(r); 420 } 421 422 /** 423 * Computes the regularised partial gamma function P. 424 * <p> 425 * See <a href="https://mathworld.wolfram.com/RegularizedGammaFunction.html">RegularisedGammaFunction</a>. 426 * Throws {@link IllegalStateException} if the iterations don't converge. 427 * @param a shape (when used as a CDF) 428 * @param x value / scale (when used as a CDF) 429 * @param epsilon Tolerance. 430 * @param maxIterations The maximum number of iterations. 431 * @return P(a,x). 432 */ 433 public static double regularizedGammaP(int a, 434 double x, 435 double epsilon, 436 int maxIterations) { 437 if (Double.isNaN(x) || (a <= 0) || (x < 0.0)) { 438 return Double.NaN; 439 } else if (x == 0.0) { 440 return 0.0; 441 } else { 442 int i; 443 double ithElement = 1.0 / a; 444 double accumulator = ithElement; 445 for (i = 1; i < maxIterations && Math.abs(ithElement/accumulator) > epsilon; i++) { 446 ithElement *= x / (a + i); 447 accumulator += ithElement; 448 if (Double.isInfinite(accumulator)) { 449 return 1.0; 450 } 451 } 452 if (i >= maxIterations) { 453 throw new IllegalStateException("Exceeded maximum number of iterations " + maxIterations); 454 } else { 455 return Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * accumulator; 456 } 457 } 458 } 459}